Method for modelling fluid displacements in a porous environment taking into account hysteresis effects

ABSTRACT

A modelling method having application to petroleum production, soil cleaning, etc. for optimizing faster and more realistically the displacement conditions, in a porous medium wettable by a first fluid (water for example), of a mixture of fluids including this wetting fluid, another, non-wetting fluid (oil for example) and a gas. The method comprises experimental determination of the variation curve of the capillary pressure in the pores as a function of the saturation in the liquid phases, modelling the pores of the porous medium by means of a distribution of capillaries with a fractal distribution by considering, in the case of a three-phase water (wetting fluid)-oil-gas mixture for example, a stratification of the constituents in the pores, with the water in contact with the walls, the gas in the center and the oil forming an intercalary layer, determination, from this capillary pressure curve, of the fractal dimension values corresponding to a series of given values of the saturation in the liquid phase, modelling the hysteresis effects that modify the mobile saturations of the fluids effectively displaced in the sample, that vary during drainage and imbibition cycles.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for modelling two-phase orthree-phase flows in a porous medium, in drainage and imbibition. It isbased on a fractal representation of the porous medium and on anoriginal approach for handling phenomena linked with hysteresis (changein the direction of variation of the saturations).

2. Description of the Prior Art

1) Experimental Studies

Experimental determination of the relative permeabilities of a porousmedium wherein a multiphase fluid flows is not easy. Measuringoperations are usually simplified by considering that one of the phasesis immobile in a state of irreducible saturation.

The values are for example acquired by means of a well-knownexperimental method referred to as “steady state” for determiningrelative permeabilities, which allows a three-phase fluid to flow withimposed flow rates between the phases. The relative permeabilitiesexpressed as a function of the two saturations are calculated byapplying Darcy's law to each phase. It is not an established fact thatthe relative permeability measurements obtained by means of this methodare really representative of the fluid displacements and, in any case,they take a long time because, at each regime change, one has to waitfor a state of equilibrium.

Another known method carries out laboratory tests in order to determinemeasurement tables (as shown in FIG. 1) relating the relativepermeabilities and the saturations for each pair of fluids of thethree-phase mixture. By adjusting experimental production curves, onetries to progressively adjust the three-phase relative permeabilities.These data tables are then entered into an Athos® type numericalsimulator which computes the fluid productions. This method is based onthe prior acquisition of many experimental measurements progressivelyadjusted by calibration and takes a long time.

2) Relative Permeability Models

The known empirical model referred to as Stone's model allows, byempirical correlations, to predict data relative to a three-phase flowfrom data corresponding to a two-phase flow. It is valid only in case ofa high water wettability and it is generally considered to be a poorpredictor.

There are two known types of physical models for modelling three-phaseflows, based on capillary pressure curves. The capillary pressure curvesare connected with a saturation (for example that of the mercuryinjected) and a pore radius from which the mercury stops for a giveninjection pressure, determined by Laplace's law, Pinj.

${{Pinj}.} = {\frac{2\sigma}{r}.}$

A first porous media representation model is described by:

Burdine, N. T.: “Relative Permeability Calculations from Pore SizeDistribution Data”, Trans AIME (1953), Vol. 198, or by

Corey, A. T.: “The Interrelation between Oil and Gas RelativePermeabilities”, Prod. Monthly (1954), Vol. 19, 38.

According to this model, the porous medium is represented by a bundle ofcylindrical capillaries with a radius distribution given by thecapillary pressure curve obtained by mercury injection. Thepermeabilities are obtained by applying Poiseuille's law to the flow offluids in these capillaries.

This model is based on the representation of the porous medium as anassembly of capillaries with different radii. The relation between thevolume and the radius of the pores is given by the value of the slope ofthe pseudo-plateau. The three fluids are supposed to share thecapillaries between them, the wetting fluid (water) occupying thesmallest ones, the least wetting fluid (gas) the largest ones and thethird fluid (oil) a zone with intermediate-size pores. It is notpossible to describe the interactions between the fluids because, insuch a model, they flow through separate channels. Finally, this modelcan be useful only if the pseudo-plateau covers a wide range ofsaturations. According to this model, the three phases of a three-phaseflow move in different capillaries and there is no interaction betweenthem.

Another known physical porous medium representation model is describedby:

de Gennes, P. G.: “Partial Filling of a Fractal Structure by a WettingFluid”, Physics of Disordered Materials 227–241, New York Plenum Pub.Corp. (1985), taken up by

Lenormand, R.: “Gravity Assisted Inert Gas Injection: MicromodelExperiments and Model based on Fractal Roughness”, The European Oil andGas Conference Altavilla Milica, Palermo, Sicily (1990).

According to this model, the inner surface of the pores is consideredisotropic and has a fractal character, and it can be modelled as a“bunch” of parallel capillary grooves so that the pores exhibit afractal cross section. The cross section of each pore is constructedaccording to an iterative process (FIG. 1). The half-perimeter of acircle of radius R₀ is divided into η parts and each of these η parts isreplaced by a semi-circle or groove. At each stage k of the process,N_(k) new semi-circular grooves of radius R_(k) and of total sectionA_(k) are created.

The fractal dimension DL of the cross section at the end of stage k isrelated to the number of objects N_(k) generated with the given scaleI_(k) by the relation:N _(k) ∞I _(k) ^(−D) ^(L)

The fractal dimension can be deduced from a mercury capillary pressurecurve according to the following procedure. Mercury is injected into aporous medium with an injection pressure that increases in stages.Laplace's law allows deducing the pore volume, knowing the volume ofmercury injected for a given capillary pressure and the drainagecapillary pressure curve relating the injection pressure to the amountof mercury injected and the curve relating the proportion of the totalvolume occupied by the pores and the size of the pores can beconstructed. In cases where a wetting liquid is drained from the porousmedium such as water by gas injection, the correlation between thegas-water capillary pressure and the saturation of the wetting phase isgiven by:

$P_{C} = S_{W}^{\frac{1}{D_{L} - 2}}$

The experimental results readily show that the values of the gas-waterrelative permeabilities expressed as a function of the threesaturations, obtained from the expressions given by the known models andthe phase distribution modes in the structure of the pores, are far fromthe measured values and therefore that the models concerned prove to betoo simplistic to represent the complex interactions that take placebetween the fluid phases.

French Patent 2,772,483 (U.S. Pat. No. 6,021,662) describes a modellingmethod for optimizing faster and more realistically the flow conditions,in a porous medium wettable by a first fluid (water for example), of amixture of fluids including this wetting fluid and at least anotherfluid (oil and possibly gas). This method involves modelling the poresof the porous medium by a distribution of capillaries with a fractaldistribution considering, in the case of a three-phase water (wettingfluid)-oil-gas mixture for example, a stratification of the constituentsin the pores, with the water in contact with the walls, the gas in thecenter and the oil forming an intercalary layer. It comprisesexperimental determination of the variation curve of the capillarypressure in the pores as a function of the saturation in the liquidphases, from which the fractal dimension values corresponding to aseries of given values of the saturation in the liquid phase arededuced. It also comprises modelling the relative permeabilitiesdirectly in a form of analytic expressions depending on the variousfractal dimension values obtained and in accordance with the stratifieddistribution of the different fluids in the pores. A porous mediumsimulator is used from these relative permeabilities to determine theoptimum conditions of displacement of the fluids in the porous medium.

The hysteresis phenomenon relates to the variations in the petrophysicalproperties (relative permeabilities, capillary pressure, resistivityindex, etc.) observed according to whether measurements are performedduring drainage or imbibition (these modes respectively correspond to asaturation increase and decrease of the non-wetting phase). Thisphenomenon must therefore be taken into account to obtain representativerelative permeability values.

The prior art concerning hysteresis effects in two-phase and three-phasemedia is described for example in the following publications:

Land C. S.: “Calculation of Imbibition Relative Permeability for Two andThree-Phase Flow from Rock Properties”, Trans AIME 1968, Vol. 243, 149,

Larsen J. A., Skauge A.: “Methodology for Numerical Simulation withCycle-Dependent Relative Permeability”, SPEJ, June 1998, and

Carlson F. M.: “Simulation of Relative Permeability Hysteresis to theNon-Wetting Phase”, SPE 10157, ATCE, San Antonio, Tex., 4–7 Oct. 1981.

FIG. 6 typically shows the course of the two-phase permeability curvesK_(rw) resulting from drainage up to irreducible saturation in wettingfluid (M), then imbibition up to residual saturation in non-wettingfluid (NM). The hysteresis phenomenon occurs at two levels. At equalsaturations S_(g), different numerical values are obtained and the endpoint reached is an unknown that depends on the cusp point S_(gM) fromwhich the displacement mode is changed. This phenomenon is usuallyattributed to the trapped non-wetting fluid fraction. At equalsaturations, the same quantity of mobile fluid is not obtained, whichdistorts the flow characteristics.

Practically all the models taking account of hysteresis effects involveLand's semi-empirical relation:

$\begin{matrix}{{\frac{1}{S_{gr}} - \frac{1}{S_{gi}}} = C_{L}} & (1)\end{matrix}$where C_(L) represents Land's constant. This relation relates theinitial saturation S_(gi) to the residual saturation S_(gr) innon-wetting fluid, in order to evaluate the saturations in trapped andfree non-wetting fluid. Assuming that this relation is valid whateverthe saturation, it is applied to determine the intermediate mobilefractions during displacement. In the case of two-phase flows,associating this relation with permeability models provides satisfactoryresults.

In the case of three-phase flows, the hysteresis of the relativepermeabilities K_(rg) takes on a particular form. A displacementhysteresis is experimentally observed, as it is the case with two-phaseflows (extent of the direction of variation of the saturations), as wellas a cycle hysteresis since the permeabilities depend on the saturationrecord. In FIG. 7, the relative permeability curves K_(rg) correspondingto a first drainage and imbibition cycle (D1 and I1 respectively) aredistinct from the corresponding curves (D2, I2) of a second cycle.

The relative permeability model developped by Larsen et al takes thesetwo forms of hysteresis into account. Starting from an approachcombining Stone's model in parallel with Land's formula and Carlson'sinterpolation method, an approach where only the displacement hysteresisis taken into account, Larsen et al have introduced an empiricalreduction factor that is a function of the water saturation, whichallows to approximate to the permeability reduction of the gasassociated with the cycle hysteresis.

SUMMARY OF THE INVENTION

The modelling method according to the invention allows faster optimizingand more realistically the displacement conditions, in a porous mediumwettable by a first fluid, of two-phase or three-phase mixturesincluding the first wetting fluid and at least a second, non-wettingfluid. It therefore provides operators with a more reliable tool forevaluating notably the best displacement modes of the fluids in theporous medium, in drainage and imbibition. It is based on a fractalrepresentation of the porous medium with modelling of the pores by adistribution of capillaries with a fractal section, considering astratified distribution of the fluids in the pores, the wetting fluidbeing distributed in contact with the walls and around the second fluid(or around the other two in case of a three-phase mixture).

The method according to the invention is applicable in many fields wherefluid flows in porous media are to be modelled in order to optimize theconditions of the displacement thereof in drainage and imbibition.Examples of fields of application are:

a) development of an oil reservoir and notably enhanced hydrocarbonproduction by injection of fluids, using for example alternateinjections of liquid and gas slugs (a method referred to as WAG). Itconstitutes an advantageous tool allowing reservoir engineers to studyalso well productivity and injectivity problems;

b) soil depollution and notably depollution of industrial sites byinjection of substances such as surfactants in polluted layers;

c) cleaning of filter cakes by displacement of the substances retainedtherein;

d) wood drying;

e) optimization of chemical reactions for example by displacement ofreaction products in a catalyst mass in order to increase the surfacesof contact, etc.

The method according to the invention is directly applicable byreservoir engineers in order to determine, for example, the mostsuitable enhanced recovery method to be applied to an undergroundhydrocarbon reservoir. The method can also serve within the scope ofindustrial site depollution operations for example.

The method comprises in combination:

experimental determination of the variation curve of the capillarypressure (Pc) in the pores of a sample of this porous medium in thepresence of a wetting fluid and of at least one non-wetting fluid (byinjection of mercury in a sample placed under vacuum for example);

determination, from this capillary pressure curve, of the fractaldimension values corresponding to a series of given values of thesaturation in liquids;

modelling the hysteresis effects that modify the mobile saturations ofthe fluids effectively displaced as a function of the number of drainageand imbibition cycles undergone by the sample, involving differentnon-wetting fluid trapping or untrapping constants according to whethera drainage stage or an imbibition stage is carried out;

modelling the relative permeabilities directly in the form of analyticexpressions depending on the various fractal dimension values obtained;and

entering the relative permeabilities in a porous medium simulator anddetermining, by means of this simulator, the optimum displacementconditions of the fluids of the mixture in the porous medium.

The method is applied for example for determining displacements of fluidmixtures comprising a first wetting fluid, a second, non-wetting fluidand a gas, considering a stratified distribution of the fluids in thepores, the wetting fluid spreading out in contact with the walls, thegas occupying the center of the pores and the second fluid beingdistributed in the form of an annular layer in contact with both the gasand the first fluid.

The method can notably be applied for determining, by means of areservoir simulator, the optimum characteristics of substances added towetting fluid slugs injected in a formation alternately with gas slugs,in order to displace hydrocarbons in place, or those of a fluid injectedinto the soil in order to displace polluting substances.

Modelling of phenomena by means of the present method has manyadvantages. It allows a better correspondence with the results obtainedin the laboratory because the physical phenomena are better taken intoaccount. The results of the model are therefore better in case of ascale change for example, for modelling an application in an operationfield.

The calculating time is reduced by comparison with the time requiredwhen tables are used as in the prior methods. Fractal type modelling canbetter deal with the hysteresis effects encountered when using WAG typeinjection processes.

The results of the method can furthermore be perfectly integrated intomany reservoir simulators: 3D, heterogeneous, composition simulators,etc.

Exploitation of the results by application softwares is facilitated. Itis no longer necessary to perform risky interpolations as it is the rulewhen working from discrete values of the result tables in order to drawisoperms for example.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the method according to the inventionwill be clear from reading the description hereafter of a non limitativeexample, with reference to the accompanying drawings wherein:

FIG. 1 illustrates, in the form of a table, the connections existing fora three-phase mixture between the experimentally obtained relativepermeability values of a fluid and the saturations for two of the threefluids;

FIG. 2 is a fractal representation of a pore;

FIG. 3 diagrammatically shows the distribution of the phases of athree-phase fluid in a fractal pore with the wetting fluid W in contactwith the wall, the gas phase G spread over the greatest part of thevolume of the pore (radius RG), the oil O being a layer between thewetting fluid and the gas;

FIG. 4 shows, as a function of the water saturation, the capillarypressure curve of a sample of Vosges sandstone for example, whose localslope S is used to determine the fractal dimension of the pores;

FIG. 5 shows, as a function of time, the production curves obtainedexperimentally for gas (G), for water (W) and for oil (O), compared withthe equivalent curves obtained by simulation by means of the methodaccording to the invention;

FIGS. 6 a, 6B show the effects of the displacement hysteresis affectingthe relative permeabilities of the wetting fluid and of the gas, K_(rw)and K_(rg) respectively;

FIG. 7 shows the combined effects of the displacement and cyclehysteresis observed experimentally on the gas phase;

FIG. 8 shows the key curves used for modelling the untrapping constant;

FIG. 9 shows the result obtained with the model within the scope of aWAG type injection; and

FIG. 10 shows the validation of the model on a WAG type experiment forthe recovery curves of the three phases.

DETAILED DESCRIPTION OF THE INVENTION

The method according to the invention allows determination of thethree-phase relative permeabilities of porous media by using a fractaltype model of the porous medium, on the basis of an approach describedby:

Kalaydjian, F. J-M et al.: “Three Phase Flow in Water-wet Porous MediaDetermination of Gas-oil Relative Permeabilities under Various Spreadingconditions”, 68th Ann. Tech. Conf. and Exh. of the SPE, Houston, Tex.,1993.

The method according to the invention comprises, as mentioned above,modelling the flow of the phases with distribution of the phases in thefractal structure of each pore. In the case of water and oil flowing ina water wet porous structure, the oil flows into the volume of the pore.In the case of a three-phase flow, there is a stratified distribution,the water, which is the wetting fluid, flows along the walls of thepores, the gas circulates in the volume of the pore and the oil flowsbetween the gas and the water. The saturations are calculated as therelative surface area in a cross section occupied by each of the fluids.

At equilibrium, all the grooves having a radius greater than Rk, whichis given by Laplace's law Pc=2γ/R_(k), are occupied by the gas, and thesmallest tubes by the two other fluids (water and oil). The wettingfluid saturation is thus expressed as the fraction of the area of theoccupied tubes.

Calculation of the fraction of the area of the capillaries occupied bythe water for all the radii between R_(k) and R_(∞) leads to thefollowing expression:

$S_{w} = \lbrack \frac{R_{k}}{R_{0}} \rbrack^{2 - D_{L}}$and, as Pc=2γ/R_(k), the correlation between the capillary pressure andthe saturation of the wetting phase is given by:

${Pc} = {Sw}^{\frac{1}{D_{L} - 2}}$where S_(w) is the saturation of the wetting phase.

The graphical representation of this correlation in a log—log diagram isa straight line starting from the point (S_(w), P_(C)) corresponding tothe largest capillary of the fractal structure with a radius R₀. One maysuppose that:

radius R₀, which is first invaded when mercury is injected (FIG. 4),corresponds to a saturation value of the order of 1/r=10³. Each segmentof the capillary pressure curve is a part of a line starting from R₀(assumed to be the same for all the different segments), correspondingto the aforementioned correlation P_(C), S_(w). Each line has a givenslope, a fractal linear dimension can be associated therewith. Thevalues of the slope range from −1.5 to −3.3 as shown in FIG. 4, whichleads to values of the fractal linear dimension D_(L) between 1.3 and1.7;

each domain is reached by the mercury for saturations corresponding tothe place where R₀ is found on each line.

The saturation of the two liquids when the gas is present in the pore iscalculated as explained above for a phase:

$S_{Lig} = \lbrack \frac{R_{k}}{R_{0}} \rbrack^{2 - D_{L}}$assuming that the two liquids occupy the tubes whose radii are less thanor equal to R_(k) and the gas, the centre of each pore. The oilsaturation is the relative area of the cross section of the groovesoccupied by oil, whose radius is less than or equal to R_(k).

Hysteresis Modelling

Calculation of the relative permeabilities requires determination, foreach phase, of the fraction circulating therein and consequentlysystematic estimation of the saturations corresponding to the stagnantfractions. This must be done for the two cases studied, for exampledrainage of water and oil by gas, and water imbibition.

The original feature of the modelling procedure accounts for thehysteresis directly at its origin, that is at the level of thenon-wetting phase trapping and untrapping phenomena. In FIG. 8, if thecurves could be deduced exactly by translation, this would mean thatpart of the gas has been trapped during the secondary drainage and doesnot take part in the flow. In fact, this is not the case but the factthat, at equal gas saturation, the permeability is lower during D2 thanduring I1 means that the trapped gas fraction is larger during drainage.In other words, the non-reversibility of the permeability curves can beexplained by a non-reversibility (hysteresis) between the trappingphenomenon and the untrapping phenomenon.

Land's formula (Equation 1) is thus kept, but an untrapping constant,valid during the drainage stages, different from Land's constant validduring the imbibition stages, is introduced. It thus all comes down tomodelling the evolution of the untrapping constant during the cycles.

The characteristics to be taken into account in the formulation are asfollows. It is considered that C_(P)=C_(L) during all the trappingstages because of being directly in Land's conditions of applicationwhere a single constant is enough to describe the phenomenon. WhenS_(gr) is low at the end of the drainage process, C_(D) is close toC_(P). This simply means here that the gas is easier to untrap when asmall amount thereof is present in the sample, because it can beconsidered that it is present in large pores that can be readilyreconnected during re-injection. When S_(gr) increases, C_(D) becomesgreater than C_(L), which shows a less efficient untrapping process.

For high gas saturations, a certain reversibility can be encountered,but a trapped gas fraction remains inacessible in small pores. A lowsingle curve K_(rg) is thus reached, which corresponds to a pseudotwo-phase case where the oil is no longer mobile.

According to this representation, based on experimental observations andworking hypotheses, the value of C_(D) passes through a maximum sincethe untrapping constant is equal to C_(L) when S_(gr) is low and when itis maximum, when the low-mobility curve is described. The followingexpression allows reconciliation of all the previous aspects:

$\begin{matrix}{C_{D} = {{( \frac{K_{rg}^{f} - K_{{rg}\;\min}}{K_{rg}^{D1}} )^{E}( \frac{S_{gt}}{S_{gr2}} )( {C_{DM} - C_{L}} )} + C_{L}}} & (2)\end{matrix}$

This formulation comprises several parameters:

C_(DM): it takes into account the difference between trapping anduntrapping,

C_(L): Land's constant,

K_(rg min): low-mobility curve,

E: calibration parameter,

K^(I) _(rg): value of the relative permeability to gas at the beginningof the previous imbibition,

K^(D1) _(rg): value of the relative permeability to gas on the firstdrainage curve for the gas saturation corresponding to K^(I) _(rg).

Whatever the order and the nature of the cycle considered, relation (2)and Land's relation (1) allow determination of the trapped and mobilesaturations by means of Land's formula while taking into account thethree-phase character of the hysteresis (displacement and cycles)without using empirical reduction factors.

$\begin{matrix}{S_{gf} = {\frac{1}{2}( {S_{g} - S_{gr} + \sqrt{( {S_{g} - S_{gr}} )^{2} + {\frac{4}{C}( {S_{g} - S_{gr}} )}}} )}} & (3)\end{matrix}$and S _(g) =S _(gt) +S _(gf)  (4)

C is equal to C_(L) or C_(D) according to the displacement mode,

S_(gf): free gas saturation,

S_(gt): trapped gas saturation.

Calculation of the Relative Permeabilities

A) Relative Permeabilities of Liquids

Application of Poiseuille's law to each capillary of the bundle for thephase that occupies the capillary allows to calculate the water and oilrelative permeabilities (FIG. 5).

Other experimental studies (Larsen et al) have shown that there is arelation allowing to relate the residual oil saturation during thecycles to the trapped gas fraction.S _(or)=(S _(or))_(Sgt=0) −aS _(gt)(S_(or))_(sgt=0) represents the saturation in residual oil left in placein the medium before gas is trapped.

If it is considered only the circulating fraction which contributes tothe hydraulic conductivity, the relative permeabilities for water andoil are expressed as follows:K _(ro) =K _(ro)(2Ph)[(S _(L) +S _(gt))^(β)−(S _(w)+(1−a)S _(gt)+(S_(or))_(Sgt=0))^(β)]K _(rw)═(S _(w) −R(1−a)S _(gt))^(β) −S _(wi) ^(β)

R is the reduction factor linked with the trapping of the non-wettingphase.

In these expressions, it is useful to mention that:

the irreducible water saturation S_(wi) is assumed to be stable,

the size range of capillaries occupied by the mobile oil is calculatedas the difference between the sizes of the capillaries occupied by thetwo liquids with the total liquid saturation S_(L)=S_(O)+S_(W) and thoseof the capillaries saturated in water and stagnant oil,

K_(ro)(2ph.) is the value of the oil relative permeability determined byan imbibition test with water and oil. When only the water and oilphases are present and since the sample tested is water wet, the oilwill flow through the section of the pore exactly as the gas in athree-phase flow.

B) Gas Relative Permeability

Since gas is a non-wetting phase, it occupies the central space of thepore and it spreads towards the periphery thereof as the gas saturationincreases, however without coming into contact with the solid wall (FIG.6). It is considered that the gas circulates in a single pore whoseradius R_(g) is given by the relation:R _(g) =R ₀ +R ₁ +R ₂ + . . . +R _(k),

the gas permeability being then given by:K _(rg) =K _(rg max.)(1−(S _(L) +S _(gt))^(α))⁴  (5)with

${\alpha = \frac{1}{2 - D_{L}}},$D_(L) being the linear fractal dimension of the porous medium and S_(L)the total liquid saturation equal to 1−S_(g).

The relative permeability model allowing calculating K_(rw), K_(ro) andK_(rg) is installed in a simulator such as ATHOS® or GENESYS©. Thisallows calibration of experiments carried out in the laboratory andoptimizing the conditions to be satisfied in order to displace petroleumfluids in place in a reservoir, either by gas injection or by alternateinjection of water and gas slugs (a method referred to as WAG), bytaking account of the pressure and temperature conditions prevailing atthe production depth.

Validation

The method according to the invention has been validated by means ofvarious types of experiments:

gas was injected into porous media containing water and oil, undervarious conditions. It can be seen in FIG. 5 for example that a verygood accordance is obtained between the production curves of the threephases (water, oil, gas) obtained experimentally and those predicted bythe reservoir simulator fed with the data obtained in accordance withthe method,

gas and water were also alternately injected (WAG type injection). FIG.10 shows that an excellent accordance is also obtained in this case forthe three phases throughout the experiment.

1. A modelling method for optimizing displacement conditions, in aporous medium wettable by a first wetting fluid, of a three-phasemixture of fluids including the first wetting fluid and at least asecond, non-wetting fluid, comprising: determining experimentally avariation curve of capillary pressure in pores of a sample of the porousmedium in a presence of the first wetting fluid and of the secondnon-wetting fluid; modelling the pores of the porous medium by adistribution of capillaries with a fractal section by considering astratified distribution of the fluids in the pores, the first wettingfluid spreading out in contact with walls of the pores and around atleast one other fluid; determining, from the capillary pressure curve,fractal dimension values corresponding to a series of given values ofsaturation in liquid phases; modelling hysteresis effects that modifymobile saturations of the fluids displaced in the sample according tothe number of drainage and imbibition cycles undergone by the sample,involving different non-wetting fluid trapping or untrapping constantsaccording to whether the drainage or the imbibition cycles are carriedout; modelling relative permeabilities directly in analytic expressionsdepending on different fractal dimension values which are obtained; andentering the relative permeabilities into a porous medium simulator anddetermining, by means of the simulator, optimum displacement conditionsfor the mixture of fluids in the porous medium.
 2. A method as claimedin claim 1, wherein the pores of the porous medium are modelled by adistribution of capillaries with a fractal distribution by considering astratified distribution of the fluids in the pores, the wetting fluidspreading out in contact with the walls, the gas occupying the center ofthe pores and the second fluid being distributed in the form of anannular film in contact with both the gas and the first fluid.
 3. Amethod as claimed in claim 2, wherein a reservoir simulator is used todetermine optimum characteristics of substances added to wetting fluidslugs injected in a formation alternately with gas slugs in order todisplace hydrocarbons in place.
 4. A method as claimed in claim 2,comprising using a reservoir simulator to determine optimumcharacteristics of a fluid injected into soil in order to drainpolluting substances.
 5. A method as claimed in claim 1, wherein areservoir simulator is used to determine optimum characteristics ofsubstances added to wetting fluid slugs injected in a formationalternately with gas slugs in order to displace hydrocarbons in place.6. A method as claimed in claim 1, comprising using a reservoirsimulator to determine optimum characteristics of a fluid injected intosoil in order to drain polluting substances.